1. Statistical Intervals for a Single Sample Chapter 8 continues Chapter 8B ENM 500 students reacting to yet another day of this

2. A Recap – the main result of Monday’s 8a presentation A 2-sided 100(1-  )% confidence Interval on µ, population variance unknown

3. Yet Another Example The following data is from a random sample of NBA player weights in pounds. Statisticweight Average:219.68 Maximum:285 Minimum:165 standard deviation27.15 sample size82 2-sided 95% CI: t.025,81

4. Today’s Excitement

5. The Behind-the-Scenes Probability Statement

6. Confidence Interval on Standard Deviation and Variance of a Normal Distribution

7. Chi-square Visually Note how chi-square moves out to the right as d.f.’s increase.

8. Some Observations on Chi-Square d.f.Alpha =.95 Upper Alpha =.05 Lower Ratio lower/upper 51.1511.07 9.63 103.9418.31 4.65 2010.8531.41 2.89 4026.5155.76 2.10 10077.93124.34 1.60 More observations lead to a tighter interval.

9. It is Time to Compute an Actual Confidence Interval for  The following sample of response times in hours for restoring power outages for Dayton Power and Light Company has been obtained:

10. One-Sided Confidence Bounds

11. Example 8-6

12. Confidence Interval For a Population Proportion, p – the Preliminaries

13. A Large-Sample Confidence Interval For a Population Proportion There is a requirement that np > 5 and n(1 – p) > 5 for using normal approximation to the binomial.

14. Large Sample C.I. For a Population Proportion cont’d If we approximate the unknown population parameter p by the estimate of p, we obtain the approximate C.I.

15. Example 8-7

16. Sample Size Determinations error

17. Problem 8-52 50 suspension helmets subjected to an impact test in which 18 were damaged. (a) 95% CI: (b) Sample size to reduce error to.02? (c) Sample size regardless of true value of p?

18. Problem 8-53 The Ohio Department of Transportation wishes to survey state residents to determine what proportion of the population would like to increase state highway speed limits to 75 mph. How many residents need to be surveyed to be at least 99% confident that the sample proportion is within 0.05 of the true proportion? The worst case would be for p = 0.5, thus with E = 0.05 and  = 0.01, z  /2 = z 0.005 = 2.58 we obtain a sample size of: n ~ 666 (a devilish result)

19. One-Sided Bounds Intuitively, this is like setting the lower bound to zero, or the upper bound to one. Then you lump all of the alpha probability onto the other side. Lower bound Upper bound

20. Tests of a Proportion - Example CBS News Poll. Sept. 14-16, 2007. N=706 adults nationwide. MoE ± 4 (for all adults). "Do you approve or disapprove of the way George W. Bush handled the situation with Iraq?" ApproveDisapproveUnsure %% ALL adults25705 Republican583111 Democrat6922 Independent20764

21. A 95 percent Confidence Interval on p

22. Margin Of Error Most surveys report margin of error (MoE) in a manner such as: "the results of this survey are accurate at the 95% confidence level plus or minus 3 percentage points." That is the error that can result from the process of selecting the sample. It suggests what the upper and lower bounds of the results are. Sampling Error is the calculated statistical imprecision due to interviewing a random sample instead of the entire population. The margin of error provides an estimate of how much the results of the sample may differ due to chance when compared to what would have been found if the entire population was interviewed.

23. Sampling Error Sample Size 1,000750500250100 Percentage near 10 2% 3%4%6% Percentage near 20 33459 Percentage near 30 344610 Percentage near 40 345710 Percentage near 50 345711 Percentage near 60 345710 Percentage near 70 344610 Percentage near 80 33459 Percentage near 90 22346

24. Tolerance and Prediction Intervals A prediction is that a little more tolerance during our weekly interval will be observed.

25. Prediction Interval for Future Observation The prediction interval for X n+1 will always be longer than the confidence interval for .

26. Where does that come from? use s to estimate 

27. An observation … or two - Here, note that the assumption of normality cannot be trivially granted under the umbrella of large sample size. Any time you are dealing with a single prediction, distribution is critical - Also, note that the elongation of the prediction interval comes from the use of the t distribution and the extra term under the radical.

28. Problem 8-56 (99% PI) The lower bound of the 99% prediction interval is considerably lower than the 99% confidence interval (1.108     )

29. Problem 8-64 The prediction interval bound is a lot lower than the confidence interval bound of 4.023 mm To obtain a one sided prediction interval, use t ,n-1 instead of t /2,n-1 Since we want a 95% one sided prediction interval, t /2,n-1 = t 0.05,24 = 1.711, and xbar = 4.05 s = 0.08 n = 25

30. 8-7.2 Tolerance Interval for a Normal Distribution Pages 733-734

31. Example 8-10

32. Tolerance Intervals for a Normal Distribution - Table XII in the appendix is the key to this. - Be careful to understand the meaning of this concept.

33. Tolerance Intervals for a Normal Distribution Even though 1.96 is the z value appropriate for a two-sided 95% confidence interval, you cannot claim that (xbar – 1.96s, xbar + 1.96s) contains 95% of the population. Sampling variation in x and s affect the size of this interval. If you knew  and , you could say that 95% of the population is in ( - 1.96 ,  + 1.96 ). Tolerance interval takes account of this uncertainty in parameter estimation to give us intervals that cover a certain percentage of the population with a given degree of confidence.

34. A Little More Tolerance – our capstone example To estimate the tire life resulting from a new rubber compound,16 tires are subjected to an end-of-life road test with the following result: 95% tolerance interval 95% confidence: K = 2.903 95% confidence Interval For  95% prediction Interval for x 17

35. This Was The Week That Was (TWTWTW) Next week We Hypothesize in Chapter 9 Learn about Type I and Type II errors and how likely that you will be making them.